David Bernier <david250@videotron.ca> wrote:>>The Kronecker-Weber theorem characterizes abelian extensions >of Q. >>If we look at p(X) = X^3 - 2 over Q, then according to >Wikipedia, the splitting field L of p over Q is> > Q(cuberoot(3), -1/2 +i*srqrt(3)/2)>>where -1/2 +i*srqrt(3)/2 is a non-trivial third root of unity.>>By Artin, because L is a splitting field,

>By Galois theory, the automorphisms of L fixing Q form a group>of order 6.

Right.

>By the Kronecker-Weber theorem, L isn't an abelian extension.

Right, L isn't an abelian extension of Q.

>But we have a non-abelian group of order 6 ...

Right.

>So I guess the automorphism group of L (which fix Q) is>isomorphic to S_3, the symmetric group on three objects.>>So, is this right?

Yes.

>Some automorphisms:>(a) identity>(b) complex conjugation>>Supposedly, there should be 4 more automorphisms of L leaving>Q invariant.>>Perhaps cuberoot(3) can be sent to either of>cuberoot(3)*(-1/2 +i*srqrt(3)/2), >cuberoot(3)*(-1/2 -i*srqrt(3)/2)?